Statistical Inference on Tooth Growth Dataset

Overview

The project provides a basic analysis of the ToothGrowth data in the R datasets package. The data gives the length of the odontoblast (teeth) in each of 10 guinea pigs at three different dosage levels (0.5, 1, and 2 mg) with two supplements (Vitamin C and Orange Juice). The following will occur;

• Loads the ToothGrowth data and perform some basic exploratory data analyses.
• Provide a basic summary of the data.
• Use confidence intervals and t-tests to compare tooth growth by supp and dose.
• States conclusions and any assumptions made.

Setting up environment

Necessary libraries for loading, manipulating, and plotting. Reading the ToothGrowth dataset into a data.table.

library(data.table)
library(ggplot2)
library(gridExtra)
library(dplyr)
library(printr)

data("ToothGrowth")
tooth_growth <- data.table(ToothGrowth)

Data Structure

str(tooth_growth)

## Classes 'data.table' and 'data.frame':   60 obs. of  3 variables:
##  $ len : num  4.2 11.5 7.3 5.8 6.4 10 11.2 11.2 5.2 7 ...
##  $ supp: Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dose: num  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ...
##  - attr(*, ".internal.selfref")=<externalptr>

head(tooth_growth)

Brief Summary

summary(tooth_growth)

Visualizations

Setting the dose variable as factors to make the plotting and t-tests easier. Plotting a grid of box plots for all the different combinations of len as it depends on supp and dose.

tooth_growth$dose <- as.factor(tooth_growth$dose)

plot1 <- ggplot(data = tooth_growth, aes(x = dose, y = len)) + 
    geom_boxplot(aes(fill = dose)) + 
    facet_grid(.~supp) +
    theme(legend.position = "None") +
    xlab(" ") + ylab("Length")

plot2 <- ggplot(data = tooth_growth, aes(x = supp, y = len)) +
    geom_boxplot(aes(fill = supp)) +
    facet_grid(.~dose) +
    theme(legend.position = "None") +
    xlab(" ") + ylab(" ")

plot3 <- ggplot(data = tooth_growth, aes(x = supp, y = len)) +
    geom_boxplot(aes(fill = supp)) +
    theme(legend.position = "None") +
    xlab("Supplement") + ylab(" ")

plot4 <- ggplot(data = tooth_growth, aes(x = dose, y = len)) +
    geom_boxplot(aes(fill = dose)) +
    theme(legend.position = "None") +
    xlab("Dosage (mg)") + ylab("Length")

grid.arrange(plot1, plot2, plot4, plot3, nrow = 2, ncol = 2)

Analysis

The goal of the project is to analyze the effect of the different supplements (Vitamin C and Orange Juice) at all of the dose levels (0.5, 1, 2 mg). The following is the code used to perform all the different combinations of t-tests. All of the test were performed using a \(95\%\) confidence level, with unequal variances assumed.

OJvsVC <- t.test(len ~ supp, data = tooth_growth)

my.t.test <- function(fac1,fac2) {
    t.test(tooth_growth$len[tooth_growth$dose == fac1],
           tooth_growth$len[tooth_growth$dose == fac2])
}

supp.t.test <- function(supp1,dose1,supp2,dose2) {
    sub_tooth <- filter(tooth_growth, (dose == as.numeric(dose1) & supp == supp1 ) |
				       dose == as.numeric(dose2) & supp == supp2)
    t.test(len ~ supp, data=sub_tooth)
}

same_supp.t.test <- function(supp1,dose1,dose2) {
    sub_tooth <- filter(tooth_growth, (dose == as.numeric(dose1) & supp == supp1 ) |
				       dose == as.numeric(dose2) & supp == supp1)
    t.test(len ~ dose, data=sub_tooth)
}

my_tests <- mapply(FUN = my.t.test,c(.5,.5,1.0),c(1.0,2.0,2.0),SIMPLIFY = FALSE)

d1 <- c(0.5,1.0,2.0,0.5,0.5,1.0,2.0,1.0,2.0)
d2 <- c(0.5,0.5,0.5,1.0,2.0,1.0,1.0,2.0,2.0)
supp_tests <- mapply(FUN = supp.t.test,rep("VC",9),d1,rep("OJ",9),d2,SIMPLIFY = FALSE)

d3 <- rep(c("OJ","VC"),each = 3)
d4 <- rep(c(1.0,2.0,2.0), times = 2)
d5 <- rep(c(0.5,0.5,1.0), times = 2)
same_supp_tests <- mapply(FUN = same_supp.t.test,d3,d4,d5,SIMPLIFY = FALSE)

t_table <- list(OJvsVC)
t_table <- append(t_table,my_tests)
t_table <- append(t_table,same_supp_tests)
t_table <- append(t_table,supp_tests)

t.stat = c(); df = c(); lower.CL = c(); upper.CL = c()
p.value = c(); mean.A = c(); mean.B = c()

rnames <- c("VC vs OJ",
	    "1.0 vs 0.5","2.0 vs 0.5","2.0 vs 1.0",
	    "OJ-1.0 vs 0.5","OJ-2.0 vs 0.5","OJ-2.0 vs 1.0",
	    "VC-1.0 vs 0.5","VC-2.0 vs 0.5","VC-2.0 vs 1.0",
	    "VC-0.5 vs OJ-0.5","VC-1.0 vs OJ-0.5","VC-2.0 vs OJ-0.5",
	    "VC-0.5 vs OJ-1.0","VC-0.5 vs OJ-2.0","VC-1.0 vs OJ-1.0",
	    "VC-2.0 vs OJ-1.0","VC-1.0 vs OJ-2.0","VC-2.0 vs OJ-2.0")

for (i in 1:19) {
    t.stat <- append(t.stat,c(t_table[[i]]$statistic))
    df <- append(df,c(t_table[[i]]$parameter))
    lower.CL <- append(lower.CL,c(t_table[[i]]$conf.int[1]))
    upper.CL <- append(upper.CL,c(t_table[[i]]$conf.int[2]))
    p.value <- append(p.value,c(t_table[[i]]$p.value))
    mean.B <- append(mean.B,c(t_table[[i]]$estimate[1]))
    mean.A <- append(mean.A,c(t_table[[i]]$estimate[2]))
}

mean.diff <- mean.A - mean.B

t_results <- data.frame("t.stat" = t.stat, "df" = df,
		        "lower.CL" = lower.CL, "upper.CL" = upper.CL,
		        "p.value" = p.value, "mean.A" = mean.A,
		        "mean.B" = mean.B, "mean.diff" = mean.diff,
		        row.names = rnames)

knitr::kable(round(t_results,4), caption = "Welch Two Sample t-test Results")

Final Conclusions

Based on the analysis performed we can conclude that as dose levels increase the tooth growth also increases. We can also conclude at the lower dose levels there is a statistical difference in the means of the 0.5 and 1.0 mg dose levels for OJ and VC with OJ being more effective on tooth growth, but at 2.0 mg OJ and VC has no significant difference.

Further study could be conducted to see if higher dosages (above 2.0 mg) would be more effective for VC or OJ. The mean of OJ seems to be decreasing at a higher rate than VC and can be seen in the difference of means for the dose levels. VC may be more effective than OJ at higher levels.

These assumptions are based on the following:

• The sample is representative of the population
• Independent variables were randomly assigned
• The distribution of the means is normal